According to Wiktionary, they all come from Latin "exitus", which is a participle of "exire", which literally means "to go out/outside, to exit, to leave".

I think that's pretty similar to the origins of success:

Learned borrowing from Latin successus, from succēdō (“succeed”), from sub- (“next to”) + cēdō (“go, move”). Partly displaced native Old English spēd, whence Modern English speed.

Both seem to (!) relate to finishing a task, and might've gained the positive connotation later on.

Failure is "fracaso". So you can exit and feeling good to have make it (éxito) or completely get destroyed by failure and never exit from your failure, living a life of despair and never be able to find the exit.

I believe that the semantic shift for ⟨exitus⟩ "coming out"→⟨éxito⟩ "success" went like this:

1. "coming out" (physically) →
2. "coming out of a situation" →
3. "satisfactory coming out of a situation" = "success".

The shift from #1 to #2 was already present in Latin, as figurative speech. That's attested by the Italian cognate ⟨esito⟩ /'ɛ.si.to/ "outcome, result". For reference English ⟨outcome⟩ shows a similar shift.

However, meaning #1 did survive into Spanish, and eventually got associated with the inherited doublet ⟨ejido⟩ /e.'xi.ðo/ "common land" (i.e. the land reached after leaving a property). Then when Spanish reborrowed ⟨éxito⟩ it got the more abstract and figurative meaning, as typical.

For 2→3, English shows a similar shift for ⟨success⟩, borrowed from Latin ⟨successus⟩. The Latin word means "the following", "the succeeding", and yet in English it was associated with satisfactory outcomes too. Also, the word ⟨outcome⟩ itself shows an example of 1→2 with native vocab.

Portuguese ⟨sucesso⟩ also shows the same shift but I don't know if the word is inherited or reborrowed.

Regarding Portuguese ⟨êxito⟩, I think that it's a lateral borrowing from Spanish. Mostly because of the vowel; Latin short /e/, if stressed, usually surfaces as open /ɛ/ in Portuguese, even in reborrowings - for reference check ⟨exército⟩ /e'zɛɾ.si.to/ "army" and ⟨épico⟩ /'ɛ.pi.ko/ "epic", both with /ɛ/. The closed /e/ makes sense however if the word was from Spanish, since it lacks the /ɛ/ vs. /e/ distinction.